Citation for published version (APA): Ochea, M. I. (2010). Essays on nonlinear evolutionary game dynamics Amsterdam: Thela Thesis

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1 UvA-DARE (Digital Academic Repository) Essays on nonlinear evolutionary game dynamics Ochea, M.I. Link to publication Citation for published version (APA): Ochea, M. I. (200). Essays on nonlinear evolutionary game dynamics Amsterdam: Thela Thesis General rights It is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulations If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 02 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. UvA-DARE is a service provided by the library of the University of Amsterdam ( Download date: 3 May 208

2 Chapter 3 Heterogenous Learning Rules in Cournot Games 3. Introduction Stability of the Cournot equilibrium in homogenous oligopoly has been constantly scrutinized: from the seminal work of Theocharis (960) who showed that the Cournot- Nash equilibrium of the standard linear inverse demand-linear cost model, loses stability once we allow for more than two oligopolists, through the exotic chaotic dynamics discovered by Rand (978) in speci c non-linear Cournot models and to the relatively recent work on evolutionary Cournot models be it in a stochastic (Alos-Ferrer (2004)) or deterministic framework (Droste et al. (2002)). Sources of instability vary from non-monotonicity in the reaction curves (Kopel (996)) to alternative speci cation of the quantity adjustment or expectations-formation processes (Szidarovszky et al. (2000)). While it may not appear too surprising to generate complicated behaviour when one allows for strange reaction functions, the sensitivity of the (linear) Cournot game outcome to the learning dynamics or to the interplay of di erent learning heuristics seems more intriguing given also the growing body 6

3 of experimental evidence on the heterogeneity of such learning rules. As our goal is to explore the impact of such learning rules heterogeneity on the limiting outcome in Cournot oligopolies, we will review a few existing results concerning adjustment processes in oligopoly either in a static framework (two boundedly rational players endowed with Cournot expectations playing a best-reply to each other s expectations) or in a population game (large group of oligopolists, randomly paired, with the resulting ecology of learning rules determined endogenously). Adaptive expectations are a straightforward generalization of Cournot expectations. They have been studied by Szidarovszky et al. (994) who found that, in a multi-player oligopoly game, if the expectation adjustment speed is su ciently low then the unique Cournot-Nash equilibrium is stabilized. Deschamp (975) and Thorlund-Petersen (990) investigate Fictitious Play duopolists, i.e. players choosing a quantity which is a best-reply to the empirical frequency distribution of the opponent past choices. Under certain restrictions on the cost structure (marginal costs not decreasing too rapidly) the process is shown to generate stable convergence to the Cournot solution. Milgrom and Roberts (99) prove convergence for a wide range of adaptive processes provided that the oligopoly game is of so-called Type I - as coined by Cox and Walker (998), and meaning that the reaction curves cross in such a way that the interior Cournot-Nash equilibrium is unique, i.e. there are no additional boundary equilibria; this has to do, again, with the marginal cost not decreasing too fast condition in Thorlund-Petersen (990). Turning to the evolutionary type of modelling oligopoly, Vega-Redondo (997) proved that imitation with experimentation dynamics leads away from the Cournot Nash equilibrium in a nite population of Cournot oligopolists. The proof uses techniques from perturbed Markov chain theory introduced to game theory by Kandori et al. (993) to show that the Walrasian equilibrium is the only stochastically stable See Camerer (2003) for a broad overview of the experimental games literature, Cheung and Friedman (997) for experiments on learning in speci c normal form games and Huck et al. (2002) for experimental evidence on learning dynamics in Cournot games. 62

4 state when the noise parameter goes to zero. (Alos-Ferrer (2004)) shows, in a similar stochastic setting, that Vega-Redondo (997) result is not robust to adding nite memory to the imitation rule: anything between Cournot and Walras is possible as long-run outcomes with only one period memory. Last, Droste et al. (2002) consider, in a deterministic set-up, evolutionary competition between Cournot players (endowed with freely available naive expectations) and rational or Nash players (who can perfectly predict the choice of the opponent but incur a cost for information gathering). The resulting evolutionary dynamics is governed by the replicator dynamics with noise and leads to local and global bifurcations of equilibria and even strange attractors for some parameter constellations. In this chapter, we construct an evolutionary oligopoly model where players are endowed with heterogenous learning procedures about the opponent s expected behavior and where they update these routines according to a logistic evolutionary dynamic. In contrast with Droste et al. (2002), our focus is the interplay between adaptive and ctitious play expectations, but other ecologies of rules (e.g. weighted ctitious play vs. rational or Nash expectations) are explored, as well. The choice of the Logit updating mechanism is motivated by the fact that this perturbed version of the Best-Reply dynamics, allows for an imperfect switching to the myopic best reply to the existing strategies distribution. The main nding is that in a population of ctitious (i.e best-responders to the empirical distribution of past history of the opponent s choices) and adaptive players (i.e. those who place large weight on the last period choice and heavily discard more remote past observations) the Cournot- Nash equilibrium can be destabilized and complicated dynamics arise. The chapter is organized as follows: Sections 3.2 and 3.3 brie y revisit the original Cournot analysis and the set of learning rules agents are endowed with. Section 3.4 introduces the model of evolutionary Cournot duopoly and evaluates alternative heuristics ecologies. Some concluding remarks are presented in section

5 3.2 Standard Static Cournot Analysis We consider a linear Cournot duopoly model much in line with the speci cation used in, for example, Cox and Walker (998) or Droste et al. (2002). Assume a linear inverse demand function: P = a b(q + q 2 ); a; b 0; q + q 2 a=b; (3.) and quadratic concave costs (i.e.decreasing marginal cost), C i (q i ) = cq i d 2 q2 i ; c; d 0: (3.2) In a standard maximization problem rm I chooses q such that it maximizes instantaneous pro ts taking as given the quantity choice of its competitor q 2 : and similarly for rm II, q = arg max q (P q C ) = arg max q [(a b(q + q 2 ))q (cq d 2 q2 )] (3.3) q 2 = arg max q 2 [(a b(q + q 2 ))q 2 (cq 2 d 2 q2 2)]: (3.4) This gives rm I and II s reaction curves as: q = R (q 2 ) = a c bq 2 2b d ; q 2 = R 2 (q ) = a c bq 2b d (3.5) and their intersection yields the interior Cournot-Nash equilibrium of the game: q = q2 = a 3b c d (3.6) 64

6 The question of how equilibrium is reached and, in particular, how could one of the duopolists know the strategic choice of the opponent at the moment when she is making the quantity choice decision generated much debate in the literature. Cournot (838) was already pointing to an equilibration process by which rms best-respond to the opponent s choice in the previous period (in current terminology naive expectations). However, there is a vast body of literature on whether and under what restrictions - for instance, the number of players and demand and cost function speci cations - the Cournot process converges. 3.3 Heterogenous Learning Rules 3.3. Adaptive Expectations In a boundedly rational environment, one possible reading of the reaction curves (3.5) is that each player best-responds to expectations about the other player s strategic choice. q (t + ) = R (q e 2(t + )) (3.7) q 2 (t + ) = R 2 (q e (t + )): A number of expectation formation or learning rules are encountered in the literature. Among them, particularly appealing are adaptive expectations where current expectations about a variable adapt to past realizations of that variable. Formally, expectations q2(t e + ) about the current period opponent quantity q 2 (t + ) are determined as a weighted average of last period s expectations and last period actual choice. q e (t + ) = q e (t) + ( )R (q e 2(t)) (3.8) = q e (t) + ( )q (t) 65

7 q e 2(t + ) = 2 q e 2(t) + ( 2 )R 2 (q e (t)) (3.9) = 2 q e 2(t) + ( 2 )q 2 (t) Fictitious Play Fictitious play, introduced originally as an algorithm for computing equilibrium in games (Brown (95)), asserts that each player best-responds to the empirical distribution of the opponent past record of play. In the context of Cournot quantitysetting games this boils down to the average of the opponent past played quantities where equal weight is attached to each past observation: q e 2(t + ) = t tx q 2 (k) k= or FP given recursively: q2(t e + ) = t q e t 2(t) + t q 2(t); t (3.0) Weighted Fictitious Play Standard ctitious play could be generalized to allow for discarding the remote past observations at a higher rate. Cheung and Friedman (997) use an exponentially-weighted scheme: q e 2(t + ) = q 2 (t) + t P u= + t P u q 2 (t u) u u= or, recursively: 8 < q2(t e + ) = : t q 2(t) e + q t t 2 (t); 2 [0; ) t t q e 2(t) + t q 2(t); = 66 9 = ; (3.)

8 It can be easily veri ed that Weighted Fictitious Play expectations nest, as special cases, the following types of expectations: (i) = 0 : Cournot or naive expectations/short memory; (ii) 0 < < : adaptive expectations with time varying weights/intermediate memory; (iii) = : Fictitious Play/long memory. 3.4 Evolutionary Cournot Games In this section we will consider an evolutionary version of the "static" model outlined in Section 3.2. Basically, at each time instance, two players are drawn randomly from an in nite population and matched to play a standard duopoly quantity-setting game. In order to form expectations about the other player s quantity choice a set of learning rules is available. Players switch between di erent rules based on some performance measure. Rules that perform relatively better are more likely to spread in the population and thus, fractions using each rule are expected to evolve over time according to a speci ed updating mechanism. A more sophisticated expectationformation rule will come at a cost compared to the simple rules which are assumed costless. Furthermore, this heterogeneity of the learning process adds uncertainty over the players expectation formation process. Since opponents may be of di erent types, expectations are formed about an average opponent strategy or quantity choice. A variety of such pairwise interactions and resulting evolution of the learning rules is discussed below, analytically (when possible) and via simulations Adaptive Expectations vs. Rational/Nash play We start with a generalization of Droste et al. (2002) behaviors ecology by considering a mixture of adaptive (nesting a special case of naive expectations as in Droste et al. (2002)) and rational or Nash players. The adaptive players behave 67

9 according to (3.8). Rational rms are able to correctly compute the quantity picked by an adaptive rm as well as the updated fractions of Adaptive and Rational players: their reaction function is playing a best-reply to the mixture of rules (equations (3.2), (3.3)): q (t) = R(q(t)) e = a c bqe (t) 2b d q2(t) e = ( n(t))q (t) + n(t)q 2 (t) (3.2) q 2 (t) = R(q2(t)) e = a c bqe 2(t) 2b d (3.3) ) q 2 (t) = a c b( n(t))r(qe (t)) 2b d + bn(t) (3.4) where, n(t) is the fraction of type II, rational players and q e (t); q e 2(t) are the expectations about a randomly encountered opponent, of the adaptive and rational types, respectively. The fraction of rational players n(t) updates according to the logistic mechanism with asynchronous updating (Hommes et al. (2005)). The asynchronous logistic updating has two components: an inertia component, parameterized by ; showing the fraction of players who do not (are not given the opportunity to) update and the logistic probability of updating for the fraction opportunity arises: of players for which a revision n(t + ) = n(t) + ( ) exp((u 2 (q 2 (t)) k)) exp((u (q (t)) + exp((u 2 (q 2 (t)) k)) (3.5) with U ; U 2 representing the average pro ts of an adaptive and rational player at time t, while denotes the inverse of the noise/or intensity of choice parameter and k stands for the extra-costs incurred by the agents employing the more sophisticated (here rational expectations) heuristic. Each strategy performance measure U (U 2 ) is computed as a fractions-weighted average of pro ts accrued in encounters with own 68

10 and other expectation-formation types: U = (a c)q + ( 2 d b)q 2 b(nq + ( n)q 2 )q (3.6) U 2 = (a c)q 2 + ( 2 d b)q 2 2 b(nq + ( n)q 2 )q 2 (3.7) Expectations - q e (t); q e 2(t) for the adaptive and rational type, respectively - about the quantity of an average opponent quantity, evolve according to: q e (t + ) = q e (t) + ( )(( n(t))q (t) + n(t)q 2 (t)) (3.8) q e 2(t + ) = ( n(t + ))q (t + ) + n(t + )q 2 (t + ) (3.9) Equations (3.5), (3.8) and (3.9) de ne a three-dimensional, discrete dynamical system (q(t e + ); q2(t e + ); n(t + )) = (q(t); e q2(t); e n(t)) describing both the evolution of the adaptive and rational types expectations and of the learning heuristics ecology: q e (t + ) = q e (t) + ( )(n(t)r(q e (t)) + ( n(t))r(q e 2(t))) q2(t e + ) = q(t) e + ( )(n(t)r(q(t)) e + ( n(t))r(q2(t))) e (3.20) n(t + ) = + exp((u 2 (R(q(t)); e R(q2(t)); e n(t)) U (R(q(t)); e R(q2(t)); e n(t)) k)) Using (3.2), (3.3) we can express rm/type II expectations as a function of rm I expectations about opponent quantity choice, e ectively reduce (3.20) to a twodimensional system: q(t e + ) = q(t) e + ( )(( n(t))r(q(t)) e + n(t) a c b( n(t))r(qe (t)) ) 2b d + bn(t) e ((U 2(R(q e (t));n(t)) n(t + ) = n(t) + ( ) (3.2) e ((U (R(q e(t));n(t))) + e ((U 2(R(q e(t));n(t)) k)) k)) Lemma 4 The Cournot-Nash equilibrium q = q 2 = a c 3b d in (3.6) together with 69

11 n = +exp(k) is the unique, interior, xed point of (3.2). For low values of the intensity of choice < = ln (d 3b)(+) b d+3b d the Cournot-Nash equilibrium is stable; at = a period-doubling bifurcation occurs and a stable two-cycle is born. The 2- cycle loses stability via a secondary Neimark-Sacker bifurcation when hits a second threshold = + + ( )(b d+(3b d))(+)( 2 k with a limit cycle surrounding each ) b point of the 2-cycle. x = a Proof. At the xed point q e (t + ) = q e (t) the following should hold: a c b( n(t))r(q e(t)) = R(q e 2b d+bn(t) (t))), R(q(t))) e = a c a c : But, R(x) = implies 3b d 3b d c 3b d and, thus, qe (t) = a c is the unique, 3b d interior2 rest point of (3.2) with corresponding fraction given by n = +exp(k) : It can be shown that system (3.2) may be rewritten in terms of deviations Q t = q e (t) q from the Cournot-Nash equilibrium steady state as (Q(t + ); n(t)) = F (Q(t); n(t) where: b 3b + d bn(t) Q(t + ) = Q(t) 2b d + bn(t) n(t + ) = n(t) + + exp((k (2b d) Q(t)b(3b d) ( 2 2b d+bn(t) )2 )) (3.22) This system has a xed point at (Q = 0; n = ). In order to investigate it +exp(k) local stability we rst derive the Jacobi matrix JF when evaluated at the steady state (Q ; n ) as: 2 4 (b 3b + d bn ) 0 2b d+bn : (3.23) 2 We note that, besides the interior Cournot-Nash equilibrium, which is main focus of this Chapter, the type II (Cox and Walker (998)) Cournot game considered, has two additional boundary equilibria (two extra intersections of the reaction functions on the x and y axis), namely:a = (q A ; q2 A ) = ( a c 2b d ; 0) and B = (qb ; q2 B ) = (0; a c 2b d ):One can show that, in a nonevolutionary environment with homogenous Cournot (naive) expectations, the Cournot-Nash equilibrium is unstable and all initial conditions (except for the 45 line) converge to a 2-cycle formed by the two Nash equilibria (A; B):However, these two additional boundary equilibria do not form a 2-cycle of our dynamical system. 70

12 Its eigenvalues are = 2b d+bn (b 3b + d bn ) and 2 = : We see that the xed point loses stability via a, primary, period-doubling, bifurcation when hits the unit circle: = 2b d+bn (b 3b + d bn ) =, = = k resulting symmetric 2-cycle f( Q; n); ( (d 3b)(+) ln : At the b d+3b d Q; n)g the following equalities hold: Thus, Q(t + ) = Q(t) Q(t) = Q(t + ) b 3b+d bn(t) 2b d+bn(t) =, n = ) +exp((k (2b d) Qb(3b [ d) 2 2b d+bn ]2 )) ; yielding Q = p s 2 ( ) (2b d) 2 b 2b = p r 2 ( ) (2b d) 2 b b 3b+d bn(t) 2b d+bn(t) b 3b+d bn(t) 2b d+bn(t) d b (3b d) 2b : Next, Q solves n = n + ( d 3b + d k ln ( + ) b d + 3b d k 2b d ( ) After further simpli cations, the 2-cycle f( Q; n); ( Q; n)g reads: 2 4 ( f( ( ) b ( ) b q k( ) 2(2b d) ; d b (3b d) 2b ); q k( ) d b (3b d) ; )g 2(2b d) 2b 3 5 (3.24) To investigate the stability of f( Q; n); ( the compound map G = F 2 evaluated at the two-cycle. Q; n)g we have to look at the Jacobian of JG( Q; n) = JF ( Q; n)jf (F (( Q; n)) = JF ( Q; n)jf ( Q; n): (3.25) The Jacobian matrix of F is: JF (Q; n) = 2 4 g g 2 g 2 g

13 with entries, g = (b 3b + d bn) 2b d+bn g 2 = Qb ( ) d 3b (2b d+bn) 2 g 2 = 2Qb2 ( )(d 3b) 2 (b exp g 22 = + 2Q2 b 3 ( )(d 3b) 2 (b (3.22): exp 2 d) exp k Q 2 b 2 (d 3b) 2 b 2 d (2b d+bn) 2 k Q 2 b 2 (d 3b) 2 b 2 d (2b d+bn) (2b d+bn) 2 k Q 2 b 2 (d 3b) 2 b 2 d (2b d+bn) 2 2 (2b d+bn) 3 2 d) exp k Q 2 b 2 (d 3b) 2 b 2 d (2b d+bn) 2 + The stability analysis can be simpli ed by exploiting the symmetry of the system G = F 2 ( Q; n) = F ( Q; n) = F (T ( Q; n)) 2 with the transformation matrix T given by: T = : 0 Each point of the 2-cycle is a xed point of the map G = F T and so, the 2-cycle of F is stable i the xed points of G are stable. governed by the Jacobian of F T : The stability of the 2-cycle is JG( Q; n) = JF ( Q; n)t It can be shown that, after further algebraic manipulations, the Jacobian takes the following form: with j 2 = b JG( Q; n) = 2 q 4 4 d 3b j 2 j 22 k 2 d 2b q ( + ) ( ) (3b d) (2b d) (b d + 3b d) j 22 = ( + ) ( ) ( ) (b d + 3b d) 3 5 k and 2(2b d) The matrix has a pair of complex conjugates eigenvalues as lengthy expressions of all the model parameters. Still, the determinant takes a more tractable form: det JG = ( + ) ( ) ( ) (b d + 3b d) + 2 b : k(+)( )( )(b d+3b d) 72

14 The Neimark-Sacker bifurcation condition (det JG = ) yields the intensity of choice threshold at which a second bifurcation arises, namely: = ( ) (b d + (3b d)) ( + ) ( b This completes the proof of Lemma 4. k ) Summing up, as the intensity of choice varies from low to high values the system transits from a unique steady state, through a 2-cycle to two limit cycles at = when it undergoes a Neimark-Sacker bifurcation and each of the 2-cycle branches becomes surrounded by a closed orbit as in Fig. 3.d. Panels (a)-(b) illustrate the limiting behavior of quantities picked by the two players as they become more responsive to payo di erences, while Panel (c) displays the time evolution of the proportion of the costly rational (or Nash) expectation-formation heuristic. The intuition is similar to Brock and Hommes (997): when the system is close to the Cournot-Nash equilibrium the simple, costless adaptive rule performs well enough and, due to the evolutionary switching mechanism, a larger number of players makes use of this cheap heuristic. However with a vast majority of agents using adaptive rule the CNE destabilizes and, as uctuations of increasing amplitude arise, it pays o to switch to the more involved, though costly, behavior (i.e. Nash) pays-o. Once a large majority of agents switches to the rational rule uctuations are stabilized and agents switch back to the costfree heuristic, re-initializing the cycle. Fig. 3.2a-b illustrates the primary (period-doubling) and secondary (Neimark-Sacker) bifurcations along with the "breaking of the invariant circle" route to chaos as the intensity of choice and degree of expectations adaptiveness are varied, respectively. Finally, Panels (c)-(d) show a strange attractor in the quantity-fraction of Nash players and quantity-quantity subspaces. 73

15 5.8 q q 2 7 q q (a) = 2: Time series (b) = 9: Time series 0.9 n (c) = 9: Evolution of fraction of RE (d) = 5:2: Qasiperiodic behavior Figure 3.: Cournot duopoly game with an ecology of Adaptive and Rational players. Panels (a)-(b): quantities evolution for di erent values of the intensity of choice (). The evolution of Nash players fraction is shown in Panel (c). Panel (d) displays a limit cycle, consisting of two closed curves around the two points of the unstable 2-cycle, arising from the secondary Hopf bifurcation. Game parameters set to a = 7; b = 0:8; c = 0; d = :; k = ; = 0:05; = 0:: 74

16 (a) Bifurcation diagram: q, = 0:05 (b) Bifurcation diagram: q ; = 5 (c) Strange attractor, q n (d) Strange attractor, q q 2 Figure 3.2: Cournot duopoly game with an ecology of Adaptive and Rational players. Bifurcation diagrams of the equilibrium quantity chosen by the Adaptive players (q ) with respect to the intensity of choice () (Panel (a)) and degree of expectation adaptiveness () in Panel (b). Game parameters set to a = 7; b = 0:8; c = 0; d = :; = 0:. Panels (c), (d) display projection of a strange attractor on the quantity-fraction of rational players and quantity-quantity subspaces, respectively. A strange attractor is obtained for the following parameterization: a = 7:54; b = 0:754; c = 0; d = :; k = :5; = 0:; = 7:8; = 0:5: 75

17 3.4.2 Adaptive vs. Exponentially Weighted Fictitious Play As was already pointed out, players form expectations about the strategy of the average opponent given the existing heuristics ecology (frequencies of each learning type). In this subsection, the subset available is restricted to Adaptive Expectations and Weighted Fictitious Play as introduced in subsection 3.3. Thus, the rational rule is replaced by a long-memory rule that relies only on past observed quantity information with, possibly, di erent weighting schemes for each observation. Similar to the previous sections, the more sophisticated heuristic comes at a cost k: Considering that a WFP process gives rise to a non-autonomous dynamical system we will discuss analytically the limit as t! when the system becomes autonomous. Let qae e (t + ) and qe W F P (t + ) denote the expectations formed about an average encountered opponent by an AE-player and a WFP-player, respectively. The evolution of these two learning rules, in a Cournotian strategic environment is given by the following system of di erence equations: q e AE(t + ) q e (t + ) = q e (t) + ( )q avg opp (t) q e W F P (t + ) q e 2(t + ) = q e (t) + ( )q avg opp (t) with > 3 : By letting n(t) denote the fraction of agents employing the more computationally-involved heuristic II (WFP expectations in this case) this system 3 In subsection 3.3 we have seen that close to 0 renders adaptive expectations, while approaching yields the standard ctitious play. These two parameters could also be interpreted as proxies for "memory": when they are close to the lower bound 0 the past is heavily discarded and only recent observations matter for constructing expectations about opponent choice, while as they are near the upper bound past information becomes increasingly important for future choices. Thus naive expectations [ = 0] is a short-memory rule, as the entire past, but last period, is discarded, while ctitious play [ = ] is a long-memory rule, as each past observation is equally important. 76

18 can be explicitly re-written as: q e (t + ) = q e (t) + ( )(( n(t))q (t) + n(t)q 2 (t)) (3.26) q e 2(t + ) = q e (t) + ( )(( n(t))q (t) + n(t)q 2 (t)) (3.27) The actual strategic choices of an AE- and W F P -player are simply best-responses to their expectations: q (t) = R(q(t)) e = a c bqe (t) 2b d q 2 (t) = R(q2(t)) e = a c bqe 2(t) 2b d (3.28) (3.29) The fraction of the more sophisticated W F P players evolves according to an asynchronous updating logistic mechanism (Hommes et al. (2005)): exp((u 2 (t) k)) n(t + ) = n(t) + ( ) exp(u (t)) + exp((u 2 (t) k)) ; (3.30) where U ; U 2 represent the average pro ts of an AE and W F P player at time t, and denotes the inverse of the noise/or "intensity of choice" parameter and k stands for the extra-costs incurred by the agents employing the more sophisticated (i.e. with a higher weight placed on past observations about opponent choices) W F P heuristic. The performance measures U ; U 2 are determined as in equations (3.6)-(3.7) from subsection Similarly, we derive a three-dimensional, discrete dynamical system (q(t+); e q2(t+); e n(t+)) = (q(t); e q2(t); e n(t)) describing both the evolution of the adaptive and ctitious play type expectations and of the learning heuristics ecology: q e (t + ) = q e (t) + ( )(( n(t))r(q e (t)) + n(t)r(q e 2(t))) q2(t e + ) = q2(t) e + ( )(( n(t))r(q(t)) e + n(t)r(q2(t))) e (3.3) n(t + ) = n(t) + + e ((U (R(q e(t));r(qe 2 (t));n(t)) U 2(R(q e(t));r(qe 2 (t));n(t))+k)) 77

19 Lemma 5 The (static) Cournot-Nash equilibrium quantities (q ; q 2) determined in (3.6) together with n = +exp(k) is an (interior) xed point of system (3.3). Proof. At the Cournot-Nash Equilibrium(CNE) q = q 2 = a c 3b d we have, by de nition, R(q e (t)) = q e (t) and R(q e 2(t)) = q e 2(t):Substituting into the rst two equations of (3.3) gives the required xed point equality. Then, n = +exp(k) solution of n(t + ) = n(t) when this equation is evaluated at the CNE. is the Local stability analysis It can be shown that the Jacobian matrix of (3.3) evaluated at the equilibrium (q; q2; n ) takes the following form: ( )( n )r ( )n r 0 J = SS = 6 ( )( n 4 )r + ( )n r 0 7 (3.32) where r i ) q e i = b 2b d ; r < : We know that under homogenous, naive (or adaptive with small ) expectations the CNE is unstable with a two- cycle emerging at the boundary. On the other hand, ctitious play alone (or weighted ctitious play with large weights put in remote past observations) stabilizes the CNE. Once the interplay between the cheap adaptive and costly WFP heuristics is introduced we can prove the following result: Proposition 6 A system with evolutionary switching between adaptive and weighted ctitious play expectations (i.e. 0 < < ) is destabilized when the sensitivity to payo di erence between alternative heuristics reaches a threshold P D = (r ) (+)(r+) ln and a stable two-cycle bifurcates from the interior CN steady k (+)(r+)+( r) state (q = q2 = a c ; 3b d n = +exp(k) ): Proof. For an ecology of Adaptive and Weighted Fictitious Play rules

20 (0; ); 2 (0; ); = the eigenvalues of the Jacobian matrix (3.32) are : ;2 = r + + r rn 2 rn 2p S; 3 = with S = r 2 2 (n ) 2 2r 2 2 n + r 2 2 2r 2 (n ) 2 + 2r 2 n + 2r 2 n 2r 2 + r 2 2 (n ) 2 2r 2 n + r 2 + 2r 2 n 2r 2 + 2r 4rn +2r 2r 2 n + 4rn 2r Next the period-doubling condition 2 = yields: n () = ( + )[( + r) + ( r)] 2r( ) P D = k ln (r ) ( + ) (r + ) ( + ) (r + ) + ( r) (3.33) We can also derive the intensity of choice threshold at which instability arises for an ecology of Naive vs. Weighted Fictitious Play - = 0; 2 (0; ) - as a special case of (3.33): P D = = k ln r( ) ( + ) ( + ) (r + ) For an ecology of Fictitious vs. Adaptive Players - = ; 2 (0; ) - system (3.3) is non-autonomous and we will investigate this case in a subsequent section, numerically. We notice that if there are no costs associated with the sophisticated predictor W F P (i.e. k = 0) the system is stable for any value of the intensity of choice. This is so because what trigger switching into the destabilizing adaptive rule are the costs incurred when using W F P : thus, for k = 0 there is no incentive to switch away from W F P even in tranquil or stable periods. The 2-cycle, in deviations from the Cournot -Nash steady state - ((x; y; n); (X; Y; n)) 79

21 with X = x; Y = y - solves the following system of equations: X = x + ( )(( n) a c bx 2b d + n a c by Y = y + ( )(( n) a c bx 2b d + n a c by x = X + ( )(( n) a c bx 2b d + n a c by y = Y + ( )(( n) a c bx 2b d + n a c by n = n + ) (a c)( ) 2b d 2b d ) (a c)( ) 2b d 2b d ) (a c)( ) 2b d 2b d ) (a c)( ) 2b d 2b d exp( b 2 2 (d 2b) 2 ) exp(((x y)(2a 2c 4bx 2by+dx+dy+2bnx 2bny)+k)) (3.34) Given that the two-cycle above satis es X = x; Y = y, we obtain, after some manipulations, that the ratio of the two expectation rules deviations from equilibrium prediction is given by: x y = ( )=( + ) ( )=( + ) For (an adaptive vs. ctitious play ecology) we get x > y. Thus, the uctuations of the adaptive, short-memory rule exceed in amplitude the oscillations incurred by the long memory heuristic (see Fig. 3.3ab). If we analyze the time evolution of the costly W F P fraction we observe a similar pattern as for the naive vs. Nash ecology discussed in the previous subsection. In tranquil times, when the dynamics is close to the Cournot-Nash equilibrium, the cheap adaptive rule outperforms the more involved and costly W F P rule. Thus, through the selection mechanism, almost the entire population becomes adaptive. However, a Cournot duopoly with adaptive(near naive) expectations is unstable and, as uctuations emerge, conditions are created for the costly, more sophisticated W F P rule, to take over the population and re-stabilize the system. We can also show, numerically, that the 2-cycle loses stability via a secondary Neimark-Sacker bifurcation with a quasi-periodic attractor consisting of two closed curves, arising around the unstable branches of the 2-cycle (Fig. 3.3d). Furthermore, the invariant curves break into a strange attractor if the intensity of choice is increased even further. Bifurcations diagrams with respect to di erent behavioral 80

22 parameters together with the evolution of a strange attractor are reported in Figures 3.4 and q q q q (a) = 2: Time series (a) = 7: Time series n (c) = 7: Evolution of WFP share (d) = 4:25: Phase plot Figure 3.3: Cournot duopoly game with an ecology of Adaptive and Weighted Fictitious Play behaviors. Periodic and chaotic time series of the quantities evolution for di erent values of the intensity of choice (Panels (a)-(b)); evolution of the WFP heuristic share in the population (Panel (c)). Panel (d) displays aquasi-periodic attractor in the space of quantitites chosen by the duopolists. Game parameters: a = 7; b = 0:8; c = 0; d = :; k = :; = 0:; = 0:9; = 0:: 8

23 (a) Bifurcation diagram, q ; = 0:9 (b) Bifurcation diagram, q ; = 0: (c) Bifurcation diagram, q ; = 0:[AE], = 0:9[W F P ] (d) Bifurcation diagram, e q ; = 0:[AE], = 0:9[W F P ] Figure 3.4: Cournot duopoly game with an ecology of Adaptive and Weighted Fictitious Play behaviors. Panels (a)-(d): bifurcation diagrams of the equilibrium quantity picked by the Adaptive type (q ) with respect to degree of expctation adaptiveness (), weighting parameter in the WFP rule (), intensity of choice () and costs associated with WFP (e), respectively. Game parameters: a = 5; b = 0:7; c = 0; d = :; k = :; = 0:; = 0:9; = 2; = 0:: 82

24 (a) Strange attractors, q q 2 ; = 3:5; = 0:5 (b) Strange attractors, q n; = 3:5; = 0:5 (c) Strange attractors, q q 2 ; = 0:8; = 0: (d) Strange attractors, q n; = 0:8; = 0: Figure 3.5: Cournot duopoly game with an ecology of Adaptive and Weighted Fictitious Play behaviors. Long-run strange attractors in the quantity-quantity and quantity-share of Adaptive players, for low (Panels (a)-(b)) and high (Panel (c)-(d)) degree of syncronicity () in the updating process. Game and behavioral parameters: a = 5; b = 0:7; c = 0; d = :; k = :; = 0:; = 0:9; = 0:: 83

25 3.4.4 Naive vs. Fictitious Play In this subsection we report numerical simulations of the long-run behavior of the evolutionary competition between costless naive expectations and costly ctitious play for a linear inverse demand-quadratic costs quantity-setting duopoly. This ecology combines, in a sense, two limiting results from the previous subsections: Cournot or naive expectations - as! 0 - along with standard ctitious play expectations - as! : It is known that ctitious play converges in a homogenous population of ctitious players to Cournot-Nash equilibrium while naive expectations alone generate continuous oscillations. Hence, a natural question to ask is whether the presence of ctitious players could stabilize a population of naive players. Fig. (3.6) presents numerical simulations of the competition between the two heuristics and show persistent uctuations of the naive players quantity around the Cournot-Nash equilibrium. The share of ctitious players (Fig. 3.6c) lingers close to extinction and spikes occasionally taking over almost the entire population. Similar to the previous two subsections, what is driving this results are the costs associated with the ctitious play rule: when the entire population follows the FP rule, the CNE is stabilized and so there is an incentive to switch to the cheaper rule, i.e. adaptive heuristic. As more and more players become adaptive the CNE loses stability, uctuations emerge and,thus, they create an advantageous environment for long-memory players to step in. Depending on the intensity of selection (as captured by parameter ) the uctuations could be periodic (Panel(a)) of even chaotic: see Panel (d) in Fig. 3.6 for numerical plot of the largest Lyapunov exponent. Fig. 3.7a-d depicts the evolution of such an attractor as the sensitivity to material payo s varies. In order to escape the long transient of the ctitious play dynamics, the phase portraits are plotted after skipping 00; 000 points of the series. While the ctitious player quantities stay within a small neighborhood the Cournot-Nash equilibrium, the strategies of the Cornout(naive) player display chaotic uctuations of large amplitudes. 84

26 Lyapunov exp. 7 q q 2 q q (a) = 2: Time series (b) = 6: Time series n (c) = 6: Evolution of FP share β (d). Largest Lyapunov exponent Figure 3.6: Cournot duopoly game with Naive and Fictitious Play Heuristics. Pattern of play evolution for naive and ctitious players, for di erent values of the intensity of choice in Panels (a) and (b), respectively. Panel (c) shows the time evolution of Fictitious Play fraction while Panel (d) provides numerical evidence for chaotic dynamics for large values of. Game and behavioral parameters: a = 7; b = 0:8; c = 0; d = :; k = ; = 0; = 0:: 85

27 (a) = :65 (b) = 2:22 (c) = 2:65 (d) = 5:38 Figure 3.7: Cournot duopoly game with Naive and Fictitious Play Heuristics. Panels (a)- (d): the evolution of the phase portrait into a strange attractor as the intensity of choice increases. Notice that the quantity picked by the ctitious player wanders around the Cournot-Nash equilibrium quantity while the output choice of the naive players exhibits more wide chaotic uctuations around the same CNE. Game and behavioral parameters: a = 7; b = 0:8; c = 0; d = :; e = ; = 0; = 0:: 86

28 3.5 Concluding Remarks We studied evolutionary Cournot games with heterogenous learning rules about opponent strategic quantity or price choice. The evolutionary selection of such learning heuristics is driven by a logistic-type evolutionary dynamics. While our focus is on the interplay between adaptive and ctitious play expectations, other rules ecologies(e.g. weighted ctitious play vs. rational or Nash expectations) are explored, as well. Analytical and simulation results about the behaviour of the corresponding evolutionary Cournot games, when various parameters of interest change, are presented. In a Cournotian setting, the main nding is that a population of (weighted) ctitious players (i.e best-responders to the empirical distribution of past history of opponent choices) could be destabilized away from the Cournot-Nash equilibrium play by the presence of the adaptive expectation formation rule (i.e. players who place large weight on the last period choice and heavily discard more remote past observations). The interaction between a costly weighted ctitious play and the cheap adaptive expectations yields complicated quantity dynamics. The key mechanism driving dynamics is the endogenous switching between learning rules based on realized tness: The ctitious play drive the system near the Cournot-Nash equilibrium where the simple, adaptive rule performs relatively well and, due, to its cost advantage, can invade and take over the more sophisticated predictor. However, the adaptive rule alone destabilizes the interior equilibrium and, when far from the CNE it pays o to bear the costs of the more involved rule reinitializing the evolutionary cycle. 87